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Can someone show that the limit bellow approaches 1/2? Can you also prove that it does, with out using the prime number theorem? $$ \lim_{n\to\infty} \frac{\sum\limits_{k=1}^n \Lambda(k) \frac{k}{n}\lceil\frac{n}{k}\rceil\{ \frac{n}{k} \}}{\sum\limits_{k=1}^n \Lambda(k)}=1/2$$

Where { n/k } is the fractional part of n/k , $ \lceil{n}/{k}\rceil{} $ is the ceiling function applied to n/k, and where $\Lambda(k)$ is the Von-Mangoldt function.

It seems intuitive because the first term involving the sum in the numerator is roughly 1/2, and if you replace the Von-Mangoldt function by some other elementry function the limit also approachs to somthing close to 1/2, but still im not able to prove this result, I thought perhaps someone might be able to convert this to an integral, similar to the way one can evaluate limits of this sort involving primarly the fractional part function and or ceiling function, I would greatly appreciate any help with an elementary proof.

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